Bayesian Methods to Improve The Accuracy of Differentially Private Measurements of Constrained Parameters

Formal disclosure avoidance techniques are necessary to ensure that published data can not be used to identify information about individuals. The addition of statistical noise to unpublished data can be implemented to achieve {\em differential privacy}, which provides a formal mathematical privacy guarantee. However, the infusion of noise results in data releases which are less precise than if no noise had been added, and can lead to some of the individual data points being nonsensical. Examples of this are estimates of population counts which are negative, or estimates of the ratio of counts which violate known constraints. A straightforward way to guarantee that published estimates satisfy these known constraints is to specify a statistical model and incorporate a prior on census counts and ratios which properly constrains the parameter space. We utilize rejection sampling methods for drawing samples from the posterior distribution and we show that this implementation produces estimates of population counts and ratios which maintain formal privacy, are more precise than the original unconstrained noisy measurements, and are guaranteed to satisfy prior constraints.